Weak Decays of Doubly Heavy Baryons: W-Exchange

Since the LHCb collaboration announced the observation of the doubly charmed baryon $\Xi_{cc}^{++}$, a series of studies of doubly heavy baryons have been presented. In this work, I analyse the non-leptonic weak decays of doubly heavy baryons $\Xi_{bc}$ and $\Omega_{bc}$ under the flavor $SU(3)$ symmetry. I mainly focus on the $W$-exchange diagrams, which will contribute to the decay channels with final states are light meson and light baryon. These channels would be helpful for searching for $\Xi_{bc}$ and $\Omega_{bc}$ at LHC. And these channels and relations of corresponding decay widths could be examined by the future experimental facilities such as LHC, Belle II and CEPC.


I. INTRODUCTION
Quark model [1][2][3][4] has predicted the existence of doubly heavy baryons, while experimentally search for these states has been for a long time [5][6][7][8][9][10][11]. In 2017, the LHCb collaboration has announced the observation of the doubly charmed baryon Ξ ++ cc with mass m Ξ ++ cc = (3621.40 ± 0.72 ± 0.27 ± 0.14)MeV via the weak decay Ξ ++ cc → Λ + c K − π + π + [12]. There is no doubt that this landmark discovery will make a substantial impact on both theoretical and experimental sides. Based on this, more experimental observations of the weak decays of doubly heavy baryons are expected, and in the meanwhile, more theoretical and phenomenological efforts are also needed.
This work is an extension and supplement of a series of previous work. Ref. [20] and [53] have analysed the weak decay of doubly heavy baryons Ξ cc,bc,bb and Ω cc,bc,bb respectively, and presented the decay amplitudes and relations of decay width for the semi-leptonic and nonleptonic processes. Beyond that, the W -exchange diagrams, which are not covered in the previous papers, will be considered in this work. These diagrams are mainly contribute to the channels whose final-state is a light meson and a light baryon. The contributions from the SU (3) singlet η 1 and φ are also considered. In addition, a number of relevant literatures about the weak decays of doubly and triply heavy baryons is listed here: [21,[54][55][56][57].
The rest of this paper is organized as follows: A briefly review of the presentations for various hadrons in flavor SU (3) symmetry will be presented in Sec.II. In Sec.III, I will analyse the W -exchange cases for the Ξ bc and Ω bc decays, and list the decay amplitudes for each decay modes. In Sec.IV, I will list the golden channels which can be used to discover the corresponding doubly heavy baryons. And a short summary will be presented in the last section.

II. PARTICLE MULTIPLES
In this section, I will begin with a brief review the representations for hadron multiplets under the flavor SU (3) group. Doubly heavy baryons Ξ bc and Ω bc can form an SU (3) triplet T bc = Ξ + bc (bcu), Ξ 0 bc (bcd), Ω 0 bc (bcs) T with the quantum number of total spin and parity J P = 1/2 + .
The light baryons form an SU (3) octet and a decuplet, with the expression of the octet and the decuplet which is symmetric in SU (3) flavor space: For the light mesons, pseudoscalar mesons form an SU (3) singlet M (P ) 1 = η 1 and an octet expressed as In particular, it should be noticed that there exists mixing between the flavor eigenstate (η 1 , η 8 ) T and the physical state (η, η ′ ) T if the flavor SU (3) symmetry is broken, while we do not take this into account in this work.
Similarly, the light vector meson singlet and octet can be written as M The weak decays of the mixed doubly heavy baryons Ξ bc and Ω bc can be categorized as three cases: charm and bottom quark decays, and W -exchange between the charm and bottom quarks. The quark-level transitions of b and c quark decays are with q 1,2,3 being light quarks. At hadron level, we can obtain the Ξ bc and Ω bc decay channels from Ξ cc and Ω cc decays with the replacement of T cc → T bc , T c → T b and D → B, and from Ξ bb and Ω bb decays with T bb → T bc , T b → T c and B → D. Therefore, it is easy to obtain the results for b and c quark decay channels from Ref. [20] and not necessary to repeat them here. For the W -exchange channels, the effective Hamitonian of the bc → uq transition is where q = d, s and V ub , V cq are CKM matrix elements. The transformation of the operators under the flavor SU (3) symmetry as 3 ⊗ 3 = 6 ⊕3, so the light quarks form the 3 and 6 representations. The anti-symmetric tensor H ′′ 3 and the symmetric tensor H 6 have the following nonzero components and for the transitions bc → ud and bc → us, respectively. Feynman diagrams for these processes are given in Fig.1, triplet, and have been contained in the b → q1q2q3 cases. So that will not be covered again here.
one should notice that the final states only contain light quarks. Thus at hadron level, the final state should be a light baryon and a light meson.

A. Decays into a light octet baryon and a light meson
If the light baryon in the final state is octet, the effective Hamiltonian at hadron level can be constructed as in which the coefficients a 1∼11 are SU (3) irreducible amplitudes. a 1∼9 denote the channels with final state mesons are SU (3) octet, whose Feynman diagrams correspond with the first three diagrams in Fig.1, and a 10∼11 denote the SU (3) singlet cases and relevant to the last diagram.
Decay amplitudes for different channels are obtained by expanding the above Hamiltonian and are collected in Tab.I. From these amplitudes, we can find the relations for the decay width in the SU (3) limit: It should be noted that above relations for decay widths are only applicable under the flavor SU (3) symmetry, in which the mass differences between final state hadrons have been ignored. And these results will be modified when considering the kinematic corrections of the final phase space.

B. Decays into a light decuplet baryon and a light meson
The effective Hamitonian for the decays of Ξcc and Ωcc into a light decuplet baryon and a light meson can be written as in which b1∼3 denote the cases final state mesons are SU (3) octet while b4 denotes the singlet cases. The corresponding Feynman diagrams have been exhibited in Fig.1.
Expanding the above Hamitonian, we will obtain the decay amplitudes in Tab.III, which leads to the following relations of the decay width: And by replacing the pseudoscalar mesons into corresponding vector counterparts, we can list the decay amplitudes for these channels in Tab.(IV) with the following relations for some of the decay widths:

IV. GOLDEN DECAY CHANNELS
Based on the previous results of decay amplitudes and relations of decay width, we will list the golden channels in this section, according to the following conditions: • CKM matrix elements: Note that the decay amplitudes in previous analysis contain a overall CKM factor V ub V * cd or V ub V * cs , in which the matrix element V cd ∼ 0.2 is much smaller than Vcs ∼ 1 [3,4]. Only the CKM allowed decay channels for Ξ bc and Ω bc will be considered.
• Detection efficiency: By considering the particle detectors in LHC, the detection efficiency of proton is higher than charged pion, and charged pion is higher than photon. And for the cascade decays, with the numbers of final state particles increasing, the detection efficiency of the detectors will necessarily decline. Moreover, the multibody phase space will be suppressed with the increase of final state particles.
To sum up, by considering the above factors, we list the all allowed decay channels of Ξ bc and Ω bc (include the suppressed ones) in Tab.VI. Among these channels, Ξ bc → pK or Ξ bc → ∆K → pπK might be used for searching Ξ bc , and Ω bc → Λ 0 K might be used to search for Ω bc at LHC.

V. CONCLUSIONS
In this work, I analysed the non-leptonic weak decays of